Dermal Exposure Under Occlusion and Wash-Off

Why Exposure Conditions Change Dermal Input

A dermal dose is not necessarily absorbed at the same rate under every exposure condition. Occlusion can change hydration, temperature, and contact between formulation and skin; changing application area can change the total permeating amount per unit time; and washing the surface can remove material that has not yet entered the skin. These changes affect the input to the body, not merely the amount written on the dose label.

The model here extends a surface-dose and skin-reservoir description. It keeps surface material distinct from material already in the skin, because washing can remove the former but does not instantaneously remove the latter. Its rate constants are effective parameters for a specified formulation, area, and exposure condition. Occlusion does not imply one universal direction or magnitude of change: the relevant parameters must be measured or justified for the application being modelled.

Baseline Surface–Reservoir Model

For a single application with initial surface amount \(S_0\), assume \(S(0)=S_0\), \(R(0)=0\), and \(A(0)=0\). Between interventions, assume constant \(k_p>0\) and \(k_r>0\), first-order transfer at each step, no surface loss, no degradation, and no other route of removal. These are simplifying assumptions: they make the amount leaving the surface enter the reservoir, and the amount leaving the reservoir enter the systemic circulation.

Solving the first equation gives \(S(t)=S_0e^{-k_pt}\). Substitution into the reservoir equation gives a linear differential equation. Multiplying it by the integrating factor \(e^{k_rt}\), integrating from 0 to \(t\), and using \(R(0)=0\) yields the following result when \(k_p\ne k_r\).

The second equation follows from mass balance: under these assumptions, \(d(S+R+A)/dt=0\), so the sum remains \(S_0\). If \(k_p=k_r=k\), the equal-rate limit is \(R(t)=S_0kt e^{-kt}\); the expression with \(k_r-k_p\) in its denominator must not be evaluated by direct substitution in that case.

Occlusion as a Change in Exposure Parameters

In this model, an occluded condition is represented by using parameters appropriate to occlusion, such as \(k_{p,\mathrm{occ}}\) and \(k_{r,\mathrm{occ}}\), in place of the unoccluded values. Occlusion may alter permeation, reservoir release, or both. It is not valid to assume from the word “occluded” alone that either rate constant must increase. The equations are applied separately under each condition, with its own parameter values and the same initial dose if the comparison holds dose constant.

Application Area and the Meaning of Dose

Area changes require clarity about what is held constant. If the local surface concentration is \(C_s(t)\) and the permeation flux per unit area is \(J(t)\), a simple flux representation gives a total transfer rate of \(A_{\mathrm{app}}J(t)\), where \(A_{\mathrm{app}}\) is application area. In a first-order surface model, an effective rate constant may be written \(k_p=P A_{\mathrm{app}}/V_s\), where \(P\) is an effective permeability and \(V_s\) is an effective surface donor volume. This representation assumes those quantities are meaningful and approximately constant over the interval considered.

If the total applied amount is held fixed, increasing area does not increase the initial total amount, but it may change \(k_p\) and therefore timing. If the dose per unit area is held fixed, increasing area also increases total applied amount. These are different comparisons and should not be conflated. In real applications, area can also alter local depletion, formulation thickness, contact, and the applicability of a single well-mixed surface compartment.

Partial Wash-Off as an Intervention

Represent a wash at time \(t_w\) as an instantaneous removal of a fraction \(f\) of the surface amount, where \(0\le f\le1\). Immediately before washing, write the amounts as \(S(t_w^-)\) and \(R(t_w^-)\). Immediately afterward,

The amount removed is \(fS(t_w^-)\). The skin reservoir is unchanged by this idealized surface wash; subsequent absorption continues from the retained surface amount and the pre-existing reservoir. After the wash, solve the same equations from the new starting amounts. This event-based treatment assumes the wash itself is instantaneous and does not change permeability or reservoir release. If washing changes those properties, the post-wash rate constants must also be changed.

Derivation, Diagnostics, and Model Choice

The key justification for treating these changes separately is conservation of amount. Between interventions, transfer from surface to reservoir subtracts \(k_pS\) from one compartment and adds exactly \(k_pS\) to the other. Transfer from reservoir to systemic circulation subtracts \(k_rR\) and adds exactly \(k_rR\) to cumulative absorption. Thus \(d(S+R+A)/dt=0\). At an instantaneous wash, the amount removed, \(fS(t_w^-)\), must be recorded as a loss from the system; it cannot also be counted as absorbed.

A frequent diagnostic is a negative reservoir amount or a reservoir amount greater than the available dose. Either suggests a sign error, mismatched rate constants in the formula, or an inappropriate model assumption. In particular, each condition’s reservoir formula uses its own denominator \(k_r-k_p\). For example, pairing the \(e^{-0.60t}\) surface term with a denominator for \(k_p=0.40\) h\(^{-1}\) does not solve the stated equations and can create impossible amounts. Also check limiting behavior: with positive \(k_p\) and \(k_r\), no losses, and sufficient time, \(S\) and \(R\) approach zero and \(A\) approaches \(S_0\). If either transfer rate is zero, that conclusion need not hold.

This compartment model is a practical alternative to using a constant systemic input rate or treating the whole applied dose as immediately absorbed. Those simpler approaches may be adequate when only an average exposure is needed and absorption is rapid relative to the time scale of interest, but they do not represent surface depletion, reservoir delay, or wash-off explicitly. A spatial skin model can represent gradients and multiple layers more realistically, but needs more parameters and data. The two-compartment model is limited by its first-order and well-mixed assumptions; it does not by itself describe nonlinear permeability, spatially nonuniform application, changing formulation, or uncertainty in exposure conditions.